544 Inferences About Normal Linear Models

In this equation,Qrepresents the sum of the squares ofnindependent random
variables that have normal distributions with means zero and variancesσ^2 .Thus
Q/σ^2 has aχ^2 distribution withndegrees of freedom. Each of the random variables
√
n(ˆα−α)/σand

√∑n

i=1(xi−x)

(^2) (βˆ−β)/σhas a normal distribution with zero
mean and unit variance; thus, each ofQ 1 /σ^2 andQ 2 /σ^2 has aχ^2 distribution with
1 degree of freedom. In accordance with Theorem 9.9.2 (proved in Section 9.9),
becauseQ 3 is nonnegative, we have thatQ 1 ,Q 2 ,andQ 3 are independent and that
Q 3 /σ^2 has aχ^2 distribution withn− 1 −1=n−2 degrees of freedom. That is,
nσˆ^2 /σ^2 has aχ^2 distribution withn−2 degrees of freedom.
We now extend this discussion to obtain inference for the parametersαandβ.
It follows from the above derivations that both the random variableT 1
T 1 =
[
√
n(ˆα−α)]/σ
√
Q 3 /[σ^2 (n−2)]

αˆ−α
√
σˆ^2 /(n−2)
and the random variableT 2
T 2 =
[√∑
n
i=1(xi−x)^2 (βˆ−β)
]/
σ
√
Q 3 /[σ^2 (n−2)]

βˆ−β
√
nˆσ^2 /[(n−2)
∑n
1 (xi−x)
(^2) ] (9.6.10)
have at-distribution withn−2 degrees of freedom. These facts enable us to obtain
confidence intervals forαandβ; see Exercise 9.6.5. The fact thatnˆσ^2 /σ^2 has a
χ^2 distribution withn−2 degrees of freedom provides a means of determining a
confidence interval forσ^2. These are some of the statistical inferences about the
parameters to which reference was made in the introductory remarks of this section.
Remark 9.6.2.The more discerning reader should quite properly question our
construction ofT 1 andT 2 immediately above. We know that thesquaresof the
linear forms are independent ofQ 3 =nσˆ^2 , but we do not know, at this time,
that the linear forms themselves enjoy this independence. A more general result is
obtained in Theorem 9.9.1 of Section 9.9 and the present case is a special instance.
Before considering a numerical example, we discuss a diagnostic plot for the
major assumption of Model 9.6.1.
Remark 9.6.3(Diagnostic Plot Based on Fitted Values and Residuals).The major
assumption in the model is that the random errorse 1 ,...,enare iid. In particular,
this means that the errors are not a function of thexi’s so that a plot ofeiversusα+
β(xi−x) should result in a random scatter. Since the errors and the parameters are
unknown this plot is not possible. We have estimates, though, of these quantities,
namely the residuals ˆeiand the fitted values ˆyi. A diagnostic for the assumption is
to plot the residuals versus the fitted values. This is called theresidual plot.Ifthe
plot results in a random scatter, it is an indication that the model is appropriate.
Patterns in the plot, though, are indicative of a poor model. Often in this later
case, the patterns in the plot lead to better models.